Abstract. We investigate discrete one-dimensional Schr\"odinger operators with aperiodic potentials generated by primitive invertible substitutions on a two-letter alphabet. We prove that the spectrum coincides with the set of energies having a bounded trace map orbit and show that it is a Cantor set of zero Lebesgue measure. This result confirms a suggestion arising from a study of Roberts and complements results obtained by Bovier-Ghez. As an application we present a class of models exhibiting purely singular continuous spectrum with probability one.